Critical points of solutions to quasilinear elliptic problems

Critical Points of Solutions to Elliptic Equations in Planar Domains with Corners

“…In particular, the Hessian matrix H u vanishes nowhere in Ω The proof follows closely the one presented in Lemma 2 and Corollary 1 of [1]. We also refer to [2] for a slightly more general version of Lemma 1.…”

“…We use the term semi-Morse to refer to critical points p of a smooth function v, such that the Hessian matrix H v (p) does not vanish (see [1,2] for more details).…”

“…Later on, Cabré and Chanillo [3] extended Makar-Limanov result; they proved that semistable solutions to problem (1) in convex domains with positive curvature have a unique nondegenerated critical point. For more recent results, we refer the reader to Finn (2008) [4], Arango and Gómez (2012) [1,2]. We also mention Grossi and Molle [8], Gladiali and Grossi [9], and Grecco [6] who have tackled this and some other related problems.…”

Critical points of solutions for the mean curvature equation in strictly convex and nonconvex domains

“…Proof. Due to the results of Theorem 11 in [3], we know that the critical set K of solution u is either finitely many isolated critical points, or is made up of exactly one critical Jordan curve, that is, the critical points and critical Jordan curve can't exist at the same time. And according to the results of [4], the constant mean curvature equation (1.1) has a unique radial symmetric solution u in concentric circle annulus domain Ω, so we deduce that the critical set K of solution u reduces to exactly one critical Jordan curve C, and C is a circle centered at the origin.…”

“…They deduced that the critical set is made up of finitely many isolated critical points. In 2012 Arango and Gómez [3] considered the geometric distribution of critical points of the solutions to a quasilinear elliptic equation with Dirichlet boundary condition in strictly convex and nonconvex planar domains respectively. In 2017 Deng and Liu [11] investigate the geometric stucture of interior critical points of solutions u to a quasilinear elliptic equation with nonhomogeneous Dirichlet boundary conditions in a simply connected or multiply connected domain Ω in R 2 .…”